Linear algebra: Analytic geometry problem.












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Let $p$ be a line given by the equation $x-1=frac{y}{2}=frac{z+3}{2}$, and $q$ a line with the equation $frac{x}{2}-1=y-2=frac{z+1}{2}$. If we reflect the $p$ over the plane $Pi$ we get the line $q$. What is the equation of $Pi$ and how many solutions are there?






linear-algebra geometry analytic-geometry plane-geometry reflection







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    0












    $begingroup$


    Let $p$ be a line given by the equation $x-1=frac{y}{2}=frac{z+3}{2}$, and $q$ a line with the equation $frac{x}{2}-1=y-2=frac{z+1}{2}$. If we reflect the $p$ over the plane $Pi$ we get the line $q$. What is the equation of $Pi$ and how many solutions are there?






    linear-algebra geometry analytic-geometry plane-geometry reflection







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $p$ be a line given by the equation $x-1=frac{y}{2}=frac{z+3}{2}$, and $q$ a line with the equation $frac{x}{2}-1=y-2=frac{z+1}{2}$. If we reflect the $p$ over the plane $Pi$ we get the line $q$. What is the equation of $Pi$ and how many solutions are there?






      linear-algebra geometry analytic-geometry plane-geometry reflection







      share|cite|improve this question









      $endgroup$




      Let $p$ be a line given by the equation $x-1=frac{y}{2}=frac{z+3}{2}$, and $q$ a line with the equation $frac{x}{2}-1=y-2=frac{z+1}{2}$. If we reflect the $p$ over the plane $Pi$ we get the line $q$. What is the equation of $Pi$ and how many solutions are there?






      linear-algebra geometry analytic-geometry plane-geometry reflection




      linear-algebra geometry analytic-geometry plane-geometry reflection






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 1 '18 at 17:00









      nene123nene123

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      284






















          1 Answer
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          $begingroup$

          Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:




          1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.

          2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.

          3. The line lies fully within the plane. Then the reflection is identical to the first line.

          4. There is no other case to consider, the list so far is exhaustive.


          Now reversing these, you get




          1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.

          2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.

          3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.

          4. If the lines are skew, there is no possible plane of reflection.


          I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.






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            1 Answer
            1






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            active

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            active

            oldest

            votes









            2












            $begingroup$

            Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:




            1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.

            2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.

            3. The line lies fully within the plane. Then the reflection is identical to the first line.

            4. There is no other case to consider, the list so far is exhaustive.


            Now reversing these, you get




            1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.

            2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.

            3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.

            4. If the lines are skew, there is no possible plane of reflection.


            I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:




              1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.

              2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.

              3. The line lies fully within the plane. Then the reflection is identical to the first line.

              4. There is no other case to consider, the list so far is exhaustive.


              Now reversing these, you get




              1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.

              2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.

              3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.

              4. If the lines are skew, there is no possible plane of reflection.


              I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:




                1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.

                2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.

                3. The line lies fully within the plane. Then the reflection is identical to the first line.

                4. There is no other case to consider, the list so far is exhaustive.


                Now reversing these, you get




                1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.

                2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.

                3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.

                4. If the lines are skew, there is no possible plane of reflection.


                I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.






                share|cite|improve this answer









                $endgroup$



                Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:




                1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.

                2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.

                3. The line lies fully within the plane. Then the reflection is identical to the first line.

                4. There is no other case to consider, the list so far is exhaustive.


                Now reversing these, you get




                1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.

                2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.

                3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.

                4. If the lines are skew, there is no possible plane of reflection.


                I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 1 '18 at 23:39









                MvGMvG

                30.8k449101




                30.8k449101






























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